A vector transport [1]}: \(T{\cal M}\oplus T{\cal M}\rightarrow T{\cal M}: \left(\eta _x,\xi _x\right)\rightarrow {\cal T}_{\eta _x}\xi _x\)  associated with a retraction \(R\)  is a smooth mapping
such that, for all \(x\in {\cal M}\) , \(\eta _x,\xi _x\in T_x{\cal M}\) , (i) \({\cal T}_{\eta _x}\xi _x\in T_{R_x\left(\eta _x\right)}{\cal M}\) , (ii) \({\cal T}_{0_x}\xi _x = \xi _x\) , and (iii) \({\cal T}_{\eta _x}\)  is a linear map. Here, \(\oplus \)  denotes the Whitney sum [1]}.
As shown in [1]},
\(\overline{{\cal T}^Q_{\eta _{\left[\bar{{\cal X}}\right]}}\xi _{\left[\bar{{\cal X}}\right]}}_{\bar{{\cal X}}+\bar{\eta }_{\bar{{\cal X}}}}:= {\cal P}^{\cal H}_{\bar{{\cal X}}+\bar{\eta }_{\bar{{\cal X}}}}\bar{\xi }_{\bar{{\cal X}}}\) 
